Channel Capacity Optimization Using MultipleInput MultipleOutput for a Rayleigh Channel
##plugins.themes.bootstrap3.article.main##
The use of multipleinput multipleoutput (MIMO) systems with multiple antenna elements provides an efficient solution for future wireless communication. In recent times, wireless communication has been characterized by high speed, higher data throughput, and high Bit Error Rate (BER) in a limited bandwidth. Low data speed handover, low Signal Noise Ratio (SNR), and hardware power drainage are common channel problems with wireless systems. The objective of this research seeks a way to optimize channel capacity and enhance system performance. The paper presents a comprehensive performance analysis of channel capacity under the Rayleigh fading channel using water waterfilling technique. MATLAB was used to analyze and simulate the process. Simulation results revealed that the waterfilling algorithm can effectively optimize channel capacity in the wireless communication system.
Downloads
Introduction
Today wireless communication system is faced with several channel impairments including but not limited to low bit error rates, drop calls, high power consumption such as those experienced in the third generation (3G), Long Term Evolution (LTE), worldwide interoperability for microwave access (WiMAX), and wireless fidelity (WIFI). To overcome these challenges, the MIMO communication technique was introduced to provide high quality of service, increase the data rate, improve coverage, reduce outage probabilities, and carry out another performance matrix of data communications. MIMO guaranteed spatial diversity gains to overcome smallscale fading effects by using diversity. It reduces path loss, by using array gain in the distribution of the antennas. It increases capacity by spatial multiplexing using parallel stream transmission and Spatial Division Multiple Access (SDMA); by focusing on a narrow transmission beam using a smart antenna. MIMO uses beamforming to enhance interference reduction and spatial multiplexing, and special MIMO for closeloop spatial multiplexing [1]. Orthogonal frequency division multiplexing is a popular wireless multicarrier transmission technique. The basic principle of OFDM is to split a high data rate stream into several low data rate streams so that the lower data rate can be transmitted simultaneously over a number of subcarriers (Spatial Multiplexing) as asserted by [2] and [3]. According to [4] and [5], different signals in MIMO systems are transmitted orthogonally at different positions in space and at least one position should be in a free fading dip.
The transmission energy can be managed and minimized with an optimum resource allocation technique. This resource allocation is fundamentally based on the convex optimization problem which uses the waterfilling algorithms principle [6]. In his study, [7] concluded that more power is allocated to a subchannel with better SNR, to maximize the sum of data rates in all subchannels, according to Shannon’s Gaussian capacity formula where each subchannel’s data rate is related to the power allocation. Each used subchannel’s SNR information will be fed back to the transmitter so that the water level is derived from the fading channel coefficient and the noise power.
The main objective of this work is to optimize the channel capacity of wireless communication systems. To achieve this, an iterative Waterfilling algorithm is used for the MIMO system. The performance analysis is recorded based on the simulation results of ergodic capacity, BitErrorRate (BER), outage probability, and channel gains using Binary phase shift keying (BPSK) modulation techniques. MATLAB simulation was used during the simulation process.
The rest of the paper is organized as follows: related literatures were discussed in Section 2. The research methodologies and system model were defined in Section 3. In Section 4, results and a discussion of simulation were presented. Finally, the summary and conclusion are presented in Section 5.
Related Literature
A MIMO system was inspired by papers presented by two renowned engineer scientists; Telatar and Foschini. They inferred that there is a dramatic linear increase in channel capacity with some antennas [5]–[7]. In [8], common practices of capacity enhancements were in the improvement that diversity projected in signal transmission, both in time and spatial diversity. Common assumptions in [9] consider two main factors, time division duplexing (TDD) mode, and hardware capability when examining the required processing performance within the coherence time interval in capacity enhancement. However, MIMO systems allow us to operate two distinct dimensions of a radio link; the first being the Diversity and the second being the Capacity [7]. In [7], [10], and [11], channel capacity increases as the number of antennas is increased, proportional to the smaller of the number of transmit antennas and the number of receive antennas. Signal fading between pairs of transmit and receive antenna elements are independent and identically shared. Reference [7] inferred that MIMO systems capacity can be linearly increased with the number of antennas. This according to [9] is due to its aid in diversity combining which increases SNR, thus resulting in capacity increase.
Spatial Diversity is employed to make the transmission more robust, though it does not increase data rate as inferred by studies carried out by [12] and [13]. Spatial diversity can be in receiving diversity and transmitting diversity. See illustration in Figs. 1 and 2.
In [14], an effort was made to address the precoding problem in MIMO ad hoc networks by maximizing the system’s mutual information under power constraints. A fast and distributed algorithm based on the quasiNewton method was developed to solve the optimization problem. Several studies [15]–[18] on beamforming techniques for MIMO capacity improvement of wireless communications provide a significant survey on the evolution and advancements in antenna beamforming and the setting of different requirements for indoor and outdoor communication scenarios and announce some basic concepts of beamforming. Scheduling is a traditional concept of simple “orthogonal” time division multiple access [16] which was enriched through the ideas of random access known as ALOHA [19]–[22], which studied a scheduling algorithm for minimizing the packet error probability in clustered TDMA networks. Their studies for a clustered wireless network prove that it may be used to improve channel capability. In another study, [23], it was discovered that transmissions from more than a single user cannot coexist successfully in the same time slot, therefore assumed and recommended Time Division Duplexing (TDD) for data scheduling as a duplexing mode as a key enabler for a new heterogeneous network architecture with the potential to provide ubiquitous coverage and unprecedented spectral area efficiencies. The study proposed the distributed implementation of a weighted sum of rate maximizationbased power allocation using a dual decomposition approach. Bensky [24], noted that power allocation plays a significant role in deciding MIMO capacity unlike in SISO. The fundamental question is the magnitude of power allocated to each transmit antenna.
Power allocation is an optimization problem to maximize capacity. Depending on the closed or opened loop mode, allocation is based on CSI. Intuitively we will allocate more power to channels based on performance quality.
Ideally, channels are expected to be naturally very turbulent due to physical and logical phenomena along and within which they are composed. The question of whether all the approaches that were studied in the literature as means of optimizing the channel affect its performances in resolving data distortion/contamination in wireless communication has to be addressed. In this research, we therefore analyzed the waterfilling algorithm by means of considering the channel SNR and making an assumption that $\sum {p}_{i}={p}_{T}$, which is a constraint on the channel. This is called the Constraint of Optimization problem. This optimization problem was resolved using the waterfilling algorithm. As an analogy to water in a pipe filling a tank, this algorithm distributes signal power depending on the presence of noise in the channel. More power will be distributed to channels with less noise. This optimum power allocation aids power optimization and enhances data throughput in the wireless communication channels.
Research Methodology
Wireless Communication System Model
The system model of a wireless communication system is, to consider a transmitter with N transmits antennas, and a receiver with M receives antennas. The channel can be modeled by the $\text{M}\text{}\times \text{}\text{N}$ matrix of H. The channel can be represented by the N × M matrix of the H channel. The N × 1 received signal y is presented in Fig. 3, given as
where x is the M × 1 transmitted vector and n is the N × 1 Additive White Guassian Noise (AWGN) vector, normalized so that its covariance matrix is an identity matrix.
${E}_{s}/{N}_{r}$ is equivalence to the power of each antenna and I_{N}, the identity matrix of N. In the formula, the H can be defined as
MIMO Channel Capacity
The capacity of a random MIMO channel with power constraint P_{T} can be presented as an expected value of
where Φ = E${\left\{x{x}^{H}\right\}}^{2}$ is the covariance matrix of the transmit signal vector x. Irrespective of the number of transmit antennas, the total transmit power is limited to P_{T.}
Estimating the capacity of a MIMO system the channel state information (CSI) is a critical parameter that must be determined.
Case 1:
When the transmitter does not know the channel; In this case, it is optimal to use a uniform power distribution [19]. Its ransmit covariance matrix is represented by
Uncorrelated noise can also be assumed in each receiver branch which is described by the covariance matrix given by
The mean capacity for a MIMO channel can therefore be presented as:
where ρ = ${P}_{T}/{\sigma}^{2}$ is average signaltonoise ratio (SNR) at each branch of the receiver. But by law of large numbers, the term $(1/nT)H{H}^{H}$→${I}_{nR}$ as ${n}_{T}$ get larger and ${n}_{T}$ is fixed.
MIMO channel ‘H’ can further be analyzed by taking diagonal the product matrix $H{H}^{H}$ either by eigenvalue decomposition or by singular value decomposition. Using eigenvalue decomposition, the product matrix is given as
where E is the eigenvector matrix with orthogonal column and ∧ is the diagonal matrix with the eigenvalue on the main diagonal. Using this notation, the mean channel capacity will be represented as:
By using the Singular value decomposition the product $H{H}^{H}$ on the channel matrix H is given as:
U and V are unitary matrices of left and right singular vectors respectively and Σ a diagonal matrix with singular values on the main diagonal. Capacity will be presented as:
With the diagonalization of the product matrix, the channel capacity of MIMO is seen to include unitary and diagonal matrices only. Thus, total capacity of a MIMO channel consists of the sum of parallel AWGN SISO subchannels. The number of parallel subchannels is determined using the rank of the channel matrix. This rank of the channel matrix is given as:
Applying the rank (H) recalling that the determinant of a unitary matrix is equal to unity, the channel capacity can be represented respectively as:
where ${\mathit{\lambda}}_{\mathit{i}}$, is the eigenvalues of the diagonal matrix ∧, and ${\mathit{\sigma}}_{\mathit{i}}^{\mathbf{\text{2}}}$, is the squares of the singular values of the diagonal matrix. The maximum capacity of MIMO is reached when n_{T} is received by an equal number of n_{R} antennas without interferences occurring, or where each transmitted signal is received by a separate set of receive antennas.
The capacity of the MIMO channel when channel information is unknown at the transmitter is maximized when all the eigenvalues $\phantom{\rule{thinmathspace}{0ex}}{\u22cb}_{i}={\u22cb}_{j}$ are equal. This happens when H is an orthogonal matrix [25]. When putting the constraints that each element of H has a power of unity. That is assumed to be equal, the maximized capacity is given as:
This means that the capacity of an orthogonal MIMO channel is M times the capacity of a SISO channel.
Case 2:
When the channel matrix is understood at the transmitter; Here, the maximum capacity of the MIMO channel may be achieved by employing a technique of distributing power to a channel of lesser noise [26] on the transmit covariance matrix. Thus, the capacity is represented as:
where ${\u03f5}_{i}$ may be a scalar, representing the portion of the available transmit power going into the ith subchannel. Power constraint at the transmitter can be represented as:
With a reduced range of nonzero singular values in the data rate equations, the capability of the MIMO channel is guaranteed to be reduced owing to a rankdeficient channel matrix. This situation occurs when the signals arriving at the receivers are correlated. Low correlation is not a guarantee of high capacity even though a high channel rank is necessary to obtain a high spectral efficiency in MIMO [26].
Adopted Model
The study adopted the waterfilling model. This model is comparable to gushing water into a vessel. In communication systems, it’s accustomed to denote an inspiration in system design style and observe for exploit methods on communications channels. In analogous, Pascal law, the amplifies system in communications network repeaters or receivers amplifies every channel to the desired power level to compensate the channel impairments [26]–[30].
Water Filling Algorithm
The waterfiling algorithm rule proves its optimality for channels having Additive White Gaussian Noise (AWGN) and intersymbol interference (ISI). It is a customary baseline algorithmic rule for numerous digital communications systems. The entire quantity of water filled symbolizes the power allocated (Pt) to the system. See illustration in Fig. 4. This is given by
where Pt is the power budget of MIMO system which is allocated among the different channels and H is the channel matrix of the system. Channel capability is given as:
For the ergodic channel capacity, considering the two case situations, the (3.5) and (3.6) are to be applied severally.
The algorithms follow as below:
1. Reorder the MIMO subchannel gain realization in an ascending order
2. Take the inverse of the channel gains.
3. Water filling has a nonuniform step structure due to the inverse of the channel gain.
4. ab initio, take the sum of the Total Power (Pt) and the Inverse of the channel gain. It provides the whole space in the waterfilling and inverse power gain.
5. Decide the initial water level by the formula given below by taking the average power allocated (average water level)
6. The power values of each subchannel are calculated by subtracting the inverse channel gain of each channel.
7. Discontinue the iteration process at any level where the Power allocated value becomes negative.
Here Pt = total level of power provide,
$(1/H)$ = Noise level (ie inverse of channel gain).
Outage Capacity
Another index of system performances of the data rate (channel capacity) is the outage probability. Capacity on the channel instant response is supposed to remain constant throughout the transmission of a finitelength coded block of data. If it falls below the outage capacity, there is a chance that the transmit block of data may be decoded with error, whichever committal to the writing theme (coding scheme) is utilized. The chance that the capacity is a smaller amount than the outage capability is denoted by the letter q. This is expressed by
In a finite probability (1 − q) where the channel capacity is higher than ${C}_{outage}$, it can also be represented as:
Simulation Results and Discussion
During this section, the results are presented and discussed based on the outcome.
Parameters for Simulation
This study deploys several parameters to simulate the channel capacity (data rate) of wireless communication. These parameters were used to support the areas of consideration within the MATlab codes. Table I shows the assorted parameters, names, and values.
S/N  Parameter  Values 

1  Transmitted power  [3], [2], [1], [4], [0] 
2  Transmitted power  10 
3  Capacity  4.7027 
4  Total power  5 
5  Input power  [4], [3], [2], [1], [0] 
6  Noise power  [2], [3], [4], [1], [5] 
7  Numbers of channels  5 
8  Frequency  Rayliegh 
9  Modulation  BPSK 
Simulation Results
In Fig. 5, the simulation indicates that channel capacity increases as SNR increases. From the simulation output, the normal Shannon capacity is lower compared to when the antenna orientations (SIMO, SIMO, MISO and MIMO) simulated. A higher capacity was obtained when water filling was applied to MIMO. Fig. 6 compares the capacity of the channel for close loop state and at open loop state. Fig. 7 is the channel bit error probability graph. Fig. 8 outage probability graph. Fig. 9 is the curve of capacity (bps/hz) against SNR.
Discussion
The MIMO channel capacity has so far been optimized based on the assumption that all transmit and receive antennas are used simultaneously. From the simulation results, there is a far difference in channel capacity when different arrays of antennas are used. In Fig. 5, and Fig. 6, channel improvement shown by the graph indicates that capacity increases as SNR increases. This correlates with [31], [32]. This was feasible when the spatialmultiplexing technique was suggested in an experiment [4]. Fig. 6 indicates that the closedloop system optimizes more than the openloop system at low SNR. It implies that at high SNR both cases have the same capacity despite CSI. We found that the relative performance of these techniques depends on the SNR and the relative values of N_{T} and N_{R}. The reason for the variation is that under a close loop, the transmit power is shared to a channel with less noise. This shows that a key part of the Eigen beamforming technique is choosing appropriate powers for each of the transmitted symbols. As SNR increases, there seems to be a convergence in capacity. However, the differences between the two conditions were slight. But there occurs a level shift ward in the graph showing lateral improvement in the channel. This correlate with several researches carried out by the majority referred [4], [5], [7], [13], [16], [18], [30], [31], [33]. These studies were based on variations in SNR in the systems. In their studies, techniques like; geometric programming, linear approximation, Legrand multipliers, singular value decomposition, and waterfilling were used. Fig. 7, curves downward indicating that reduced bit error rate gave high performances on the channel. This high performance resulted from a rise condition of high SNR in the channel. With the waterfilling technique employed in the channel, there is little variation in the channel performance at the cutoff point within the minimum and maximum values of the E_{b}N_{o.}
MIMO presents the best option in an outage condition. The SNR within the channel coverage areas is an optimum determinant for an outage. At reduced channel capacity, outage probability is extremely high (Fig. 8), therefore MIMO with water filling technique is most preferred. Complementary CDF is a better measurement tool to measure the performance of the channel. It enables network designers to foresee available bands, given the target data rate, depending on the quality of service (QoS) of the channels. The system capacity gain (Fig. 9), shows that gain increases steadily to a given SNR threshold and then starts decreasing steadily. However, without water filling, the gain decreases as SNR increases.
Conclusion
The performance of the wireless communication system is better when water waterfilling algorithm is applied to MIMO. This technique allocates optimum power to subchannels with less noise. These power allocation principles provide energy efficiency with respect to the corresponding average channel capacity by providing a high energy rate. The results of the simulations show that channel capacity is optimized by deploying the waterfilling technique. The simulation indicates a linear increment in capacity when the system SNR increases. There was a significant performance of about 50% performance indication between conventional MIMO and MIMO with waterfilling technique.
References

Ghosh SK, Mehedi J. Performance analysis of channel capacities in different diversity techniques. Helix. 2018;8(2):3176–9, Available from: http://www.ijdacr.com.
DOI  Google Scholar
1

Viswanathan M. Single input multiple output (SIMO) model for reciever diversity (online). September 6, 2019. Available from: www.guassianwave.com/singleinputmultipleoutput.
Google Scholar
2

Anisimov A. MIMO overview (online). December 14, 2019. Available from: Http://anisimoff.org/eng/general/MIMO.html.
Google Scholar
3

Lokesh A, Mahesh KP. Capacity estimation in MIMOOFDM system for different fading channel using water filling algorithms’ (online) (Vol. 3, Issue 3, October 2014). Available from: www.ijdacr.com.
Google Scholar
4

Telatar E. Capacity of multiantenna Gaussian channels. Eur Trans Telecommun. 1999;10(6):585–96. 12 September 2008. doi: 10.1002/ett.4460100604.
DOI  Google Scholar
5

Foschini GJ, Gans MJ. On limits of wireless communications in a fading environment when using multiple antennas. Kluw Commun. 1999;6:311–35, Retrieved November 28 2019.
DOI  Google Scholar
6

Garg V. Wireless Communications and Networking. 340 Pine Street, Sixth Floor San Francisco, CA, United States:Morgan Kaufmann Publishers Inc; 2020. pp. 840. ISBN:9780123735805.
Google Scholar
7

Abramson N. “The Aloha system”: another alternative for computer communications (online). 2014. Available: https://www.researchgate.net/publication/234790393.
Google Scholar
8

Stuber GL. Principle of Mobile Communication. 4th ed. Wiley Library online, 2017. ISBN 97833956147. www.onlinelibrary.wiley.com.
Google Scholar
9

Boyd S, Vandenberghe L. Convex Optimization with Engineering Applications. Course Notes, EE364, Stanford University Press; 1998.
Google Scholar
10

Foschini GJ. Layered spacetime architecture for wireless communication in a fading environment using multiple antennas. Bell Labs Tech J. Autumn 1996;1(2):41–59.
DOI  Google Scholar
11

Bensky A. Radio propagation. In ShortRange Wireless Communication. 3rd ed. Switzerland: Springers International Publishers, 2019. Available from: https://www.sciencedirect.com/topics/engineering/spatialdiversity.
DOI  Google Scholar
12

Arapoglou PD, PrietoCardeira R, De Gaudenzi R. MIMO over satellite. Cooperative Cogn Satellite Syst. 2015. doi: 10.1016/B9780127999487.000086.
DOI  Google Scholar
13

Bensky A. Antenna and transmission lines. In Shortrange Wireless Communication. 2nd ed. Newnes, 2003, pp. 39–74. doi: 10.1016/B97807500/500224.
DOI  Google Scholar
14

Fakih K, Diouris J, Andrieux G. Transmission strategies in MIMO Ad Hoc networks. J Wireless Commun Netw. 2009;2009:128098. doi: 10.1155/2009/128098.
DOI  Google Scholar
15

Health RWD, GomadezPrelcic N, Rangan S, Roh W, Sayeed AM. An overview of signal processing technique for mutimeter wave MIMO systems. IEEE J Sel Top Signal Process. 2016;10(3):436–53. doi: 10.1109/JSTSP.2016.2523924.
DOI  Google Scholar
16

Rakesh RT, Sen D, Das G. Beamforming for millimeter wave communications: an inclusive survey. IEEE Commun Surveys Tutorials. 2015;2016:2–9. doi: 10.1109/COMST.2015.250460018.
Google Scholar
17

Murray BP, Zaghloul AI. A survey of cognitive beamforming technique, united state national committee ofURSI national Radio science meeting. 2014. Retrieved March 2019 from: http://dx.doi.org/10.1109/USNCURSINRSM2014.
DOI  Google Scholar
18

Vouyioukas D. A survey on beamforming techniques forreless multi input multi output relay network. Int J Antenna Propagation. 2013;18(2):949–73. RetrievedMarch 2019 from: http://dx.doi.org/10.1155/2013/745018.
DOI  Google Scholar
19

Roberts LG. Aloha packet system with and without slots and capture. Newsletter ACM Sigcomm Comput Commun Rev. 1975;5(2):6–7.
DOI  Google Scholar
20

Arash T. Tayserkani, Mats Rydstrom, Erik G. Strom, Arne Svensson. Scheduling algorithms for minimizing the packet error probability in clusterized TDMA networks. EURASIP J Wirel Commun Netw. 2009;2013:9–18. Retrieved March 2019 from: https://researchgate.net/publication/22037031.
DOI  Google Scholar
21

Lee JN, Mazindar RR, Shroff NB. Joit opportunities power scheduling and endtoend rate control for wireless adhoc network. IEEE Trans Veh Technol. 2007;58920:801–9.
DOI  Google Scholar
22

Choi J, Lee N, Hong SN, Caire G. Joint user scheduling, power allocation, and precoding design for massive MIMO systems: a principal component analysis approach. Proc. IEEE Int. Symp. Inf. Theory (ISIT), pp. 396–40, Jun. 2018.
DOI  Google Scholar
23

Bensky A. Introduction to information theory and coding. In ShortRange Wireless Communication. 3rd ed. Wiley Online library. 2019. (Online). Available from: www.sciencedirect.com/engineering/, www.onlinelibrary.wiley.com.
DOI  Google Scholar
24

Das SS. Lecture Note on “Fundamentals of MIMO Wireless Communication”, Mooc. IIT Kharagpur, India: G.S. Sanyal School of Telecommunications; July 2016.
Google Scholar
25

Phan K, Long bao L, vorobyau S, LeNgoc. Power allocation and admission control in multi user relay network, via convex programing, central and distributed scheme. EURASIL J Wirel Commun Netw. 2009;56(2):801–9.
DOI  Google Scholar
26

Boyd S, Kim SJ, Vandenberghe L, Hassibe A. A Tutorial on Geometries Programming. Spriner Science Plus BusinessMedia LCL111; 2007.
DOI  Google Scholar
27

Huang CH, Kao HY. An effective linear method for geometric programming problem. IEEE Conference Publication 7431746, 2009.
DOI  Google Scholar
28

Aritra D, Tirthanka D. Improving MIMO performance using water filling Algorithm. IJEDR vol. 4, issues 2, 2016. ISDN 23219939. “Iterative DynamicWaterfilling for Fading MultipleAccessChannels with Energy Harvesting”, IEEEAT & T LabsResearch, 2014. Retieved on August 20, 2020 from: https://www.researchgate.net/publication/259657591.
Google Scholar
29

Kobayashi M, Caire G, Bingulac P. An iterative waterfilling algorithm for maximum weighted sumrate of Gaussian MIMOBC, IEEES. EURASIP Journal on Wireless Communication and Networks, 2006. April 2009 Edition. doi: 10.1109/JSAC.2006.879410.
DOI  Google Scholar
30

Wang Z, Aggarwal V, Wang X. Iterative Dynamic Waterfilling for Fading MultipleAccess Channels with Energy Harvesting. IEEEAT&T LabsResearch; 2014. Retieved on August 20, 2020 from: https://www.researchgate.net/publication/259657591.
Google Scholar
31

Zaidi A, Athley F, Medbo J, Gustavsson U, Durisi G, Chen X. 5G physical layer. In Principles, Models and Technology Components. Academic Press Library, 2018. ISBN: 9780128145784.
Google Scholar
32

Foschini GJ, Gans MJ. On limits of wireless communications in a fading environment when using multiple. 1999;8(2007):67–127.
Google Scholar
33